In this section we use the work done in the previous section, based on [1]} and [2]}, to address the \(L^{r}\) -propagation theory filling along the way some of the issues that remained open in [3]}. We free the scattering kernel \(b\)  of any higher integrability assumption, the only requirement is \(b\in L^{1}(\mathbb {S}^{d-1})\) , and deal with the case \(r=\infty \)  as well. In addition, we present a quite natural argument to find a lower bound for the negative collision operator to clearly address this issue with a minimal requirement on solutions. For a brief comparison between cutoff and non-cutoff \(L^r\) -propagation theories we refer the reader to [4]}.
